Linear Quaternion Differential Equations: Basic Theory and Fundamental Results

Quaternion-valued differential equations (QDEs) is a new kind of differential equations which have many applications in physics and life sciences. The largest difference between QDEs and ODEs is the algebraic structure. On the non-commutativity of the quaternion algebra, the algebraic structure of the solutions to the QDEs is completely different from ODEs. It is actually a left- or right- free module, not a linear vector space. This paper establishes a systematic frame work for the theory of linear QDEs, which can be applied to quantum mechanics, fluid mechanics, Frenet frame in differential geometry, kinematic modelling, attitude dynamics, Kalman filter design and spatial rigid body dynamics, etc. We prove that the algebraic structure of the solutions to the QDEs is actually a left- or right- module, not a linear vector space. On the non-commutativity of the quaternion algebra, many concepts and properties for the ordinary differential equations (ODEs) can not be used. They should be redefined accordingly. A definition of Wronskian is introduced under the framework of quaternions which is different from standard one in the ordinary differential equations. Liouville formula for QDEs is given. Also, it is necessary to treat the eigenvalue problems with left- and right-sides, accordingly. Upon these, we studied the solutions to the linear QDEs. Furthermore, we present two algorithms to evaluate the fundamental matrix. Some concrete examples are given to show the feasibility of the obtained algorithms. Finally, a conclusion and discussion ends the paper.